Hi there,
I am an expert in JavaScript with more than two years of experience. I can create an installable trigger in your google sheet that can act on document changes (when you add a new URL for instance). Following the event, the associated handler can then send the URL to a URL shortening service and get a shortened URL in response. There are many APIs and most requires payment, API token or both. There are also limits on how many calls can be made to the API per second. So, any suitable service (suitable for your use case) can be integrated.
To get started, I will need you to find a URL shortening service and create an account and obtain developer keys for it. Then I will create a mock document with the trigger which, I will share with you so you can check for yourself. If it's working as expected, I will send the trigger code or install it on your sheets file (whichever approach suits you best).
Hope to hear from you soon. Thanks for your time and consideration.